Determine the second partial sum of the power series solution

I have been given an hypergeometric equation looking like this $$x y''(x)+(4-x)y'(x)+\frac{7}{2}y(x)=0$$

I have found the power series solution to the above equation: $$\frac{7}{2}a_0+8a_1+\sum_{n=1}^{\infty}\left(a_{n+1}n(n+1)+4a_{n+1}(n+1)-a_n n+\frac{7}{2}a_n\right)x^n$$

And I have also found the recurrence formula for the equation: $$\frac{a_{n+1}}{a_n}=\frac{-\frac{7}{2}+n}{(n+1)(n+4)}$$

My problem is that I have to state the 2nd partial sum of the power series solution found for the hypergeometric equation with the initial condition y(0) = 2. I have also been told that the 2nd partial sum can be written as: $$y_2(x)=\sum_{n=0}^{2}a_nx^n,y_2(0)=2$$ Can someone help me with this? Because I am lost.

• What is $y_2$ Is it $y$? Nov 25 at 10:16
• It is how it's stated in the exercise. I think there is a spelling mistake. But $$y_2$$ is supposed to represent the 2nd partial sum of the power series solution Nov 25 at 10:21